Nonlinear Dynamic Surface Control of Chaos in Permanent Magnet Synchronous Motor Based on the Minimum Weights of RBF Neural Network

Abstract

This paper is concerned with the problem of the nonlinear dynamic surface control (DSC) of chaos based on the minimum weights of RBF neural network for the permanent magnet synchronous motor system (PMSM) wherein the unknown parameters, disturbances, and chaos are presented. RBF neural network is used to approximate the nonlinearities and an adaptive law is employed to estimate unknown parameters. Then, a simple and effective controller is designed by introducing dynamic surface control technique on the basis of first-order filters. Asymptotically tracking stability in the sense of uniformly ultimate boundedness is achieved in a short time. Finally, the performance of the proposed controller is testified through simulation results.

1. Introduction

The permanent magnet synchronous motor is widely used in the industrial applications [1, 2]. But PMSM with nonuniform breath appears to be the chaotic behavior at specific parameters and working conditions. This behavior leads to the intermittent oscillation of torque and speed, irregular current noise of the system, and unstable control performance. Furthermore, it intensively influences the stability and safety of the system [3].

For ameliorating the performance of the PMSM system, a large amount of literatures and control methods have been attempted to apply to the motor. For example, to improve the error convergence rate, the nonsingular fast terminal sliding mode control (SMC) [4] which can reach finite-time stability is applied. In [5], a high-order SMC method via backstepping is presented to attain finite-time tracking control regardless of mismatched disturbance. However, the controller appears to be discontinuous phenomenon in dynamic sliding manifolds. In [6], an adaptive backstepping based terminal sliding mode controller for parameter strict-feedback system is proposed, where the finite-time convergence of the error is achieved.

Furthermore, the OGY method is a fundamental technology for controlling chaos [7, 8]. Unfortunately, choosing an adjustable parameter usually becomes very difficult in real practice. Control of chaos by using the time-delay feedback control technology is introduced to the real applications, but it suffers from some problems as the control objective must be the equilibrium [9]. The dynamic surface control developed by Swaroop et al. [10] is a control technique by introducing a filter at each recursive step of the backstepping design procedure, so the differentiation items on the virtual function can be avoided. DSC has been pioneered by the work of Swaroop et al. about 10 years ago, but it does not consider nonlinear plant with uncertain time delays and disturbances. Also, nonlinear items are assumed to be completely known and the control gain is equal to one. In 2005, by incorporating DSC into a neural network based adaptive control design framework, Wang and Huang proposed a backstepping based control design for a class of nonlinear systems in strict-feedback form with arbitrary uncertainty [11]. However, uncertain time delays and disturbances are not involved in the model and the control gain is equal to one. Zhang et al. in 2008 proposed adaptive DSC for a class of pure-feedback nonlinear systems with unknown dead zone and perturbed uncertainties using neural networks [12], Na et al. in 2011 [13] presented adaptive neural dynamic surface control for servo systems with unknown dead zone, Li et al. in 2013 presented an adaptive fuzzy DSC output feedback approach for a single-link robotic manipulator coupled to a brushed direct current motor with a nonrigid joint [14], and Tong et al. in 2013 presented an adaptive fuzzy decentralized backstepping output feedback control approach for a class of uncertain large-scale stochastic nonlinear systems without the measurements of the states [15]. In their works, it is assumed that there exists positive constant which satisfies specified constraint condition. But it is very difficult to define the boundedness of unknown control gain in real practice. Incidentally, the PMSM is known to exhibit chaotic behavior under certain conditions. Whether the latter methods can suppress the chaos oscillation in PMSM needs further research since DSC with RBF has been pioneered by the work of Wang et al. In addition, Hu et al. overcame the gain-constrained recursive filtering challenge for a class of time-varying nonlinear stochastic systems with probabilistic sensor delays and correlated noises [16] and solved the recursive filtering problem for a class of discrete-time nonlinear stochastic systems with random parameter matrices, multiple fading measurements, and correlated noises [17]. He further investigated the recursive finite-horizon filtering problem for a class of nonlinear time-varying systems subject to multiplicative noises, missing measurements, and quantisation effects [18, 19]. The issues discussed in these literatures offer significant references to the research of chaos control in PMSM.

Inspired by the work above, a new approach to design the nonlinear dynamic surface controller based on the minimum weights of RBF neural network is proposed for permanent magnet synchronous motor with the unknown parameters, disturbances, and chaos. During the controller design process, RBF neural network is employed to approximate unknown nonlinear functions. The main difficulty encountered in the controller design process is how to deal with the unknown control gain in the system. To overcome this difficulty, the adaptive method was also introduced to handle the problem. The proposed controller guarantees a good tracking performance and the boundedness of all the signals in the closed-loop system. Furthermore, the suggested controller contains the minimum weights of RBF neural network. As a result, the computational burden of the scheme is greatly alleviated. This makes our design controller more suitable for practical applications.

2. Mathematic Model

It is well known that the PMSM is applied widely in the motor drives, servo systems, and household appliances owing to advantages, for instance, simple structure, high efficiency, high power density, and low manufacturing cost [20]. However, the PMSM is experiencing chaotic behavior when the system parameters are falling into a special area, which can lead to the enormous destruction. The mathematic model of the PMSM based on the 𝑑-𝑞 axis is given as follows [21, 22]: 𝑑𝜔𝑑𝑡=𝑛_𝑝𝐽𝐿_𝑑−𝐿_𝑞𝑖_𝑑𝑖_𝑞+𝑛_𝑝𝐽𝜓_𝑟𝑖_𝑞−𝐵𝐽1𝜔−𝐽𝑇_𝐿,𝑑𝑖_𝑞𝑑𝑡𝑅=−𝐿_𝑞𝑖_𝑞−𝐿_𝑑𝐿_𝑞𝜔𝑖_𝑑−𝜓_𝑟𝐿_𝑞1𝜔+𝐿_𝑞𝑢_𝑞,𝑑𝑖_𝑑𝑑𝑡𝑅=−𝐿_𝑑𝑖_𝑑+𝐿_𝑞𝐿_𝑑𝜔𝑖_𝑞+1𝐿_𝑑𝑢_𝑑. (1)

The denotations of the PMSM parameters are shown in Table 1. Suppose the direct and the quadrature-axis winding inductances are equal; that is, 𝐿=𝐿_𝑑=𝐿_𝑞. Meanwhile assume the time scale 𝜏 to be such that 𝜏=𝐿/𝑅 and define the normalized time 𝑡 to be such that 𝑡=𝑡/𝜏 and the scalar 𝜅 to be such that 𝜅=𝐵/(𝑛_𝑝𝜏𝜓_𝑟). Finally, the scaled state variables 𝜔, 𝑖_𝑑, and 𝑖_𝑞 are defined as follows: 𝜔=𝜔𝜏,𝑖_𝑑=𝑖_𝑑𝜅,𝑖_𝑞=𝑖_𝑞𝜅, (2) where 𝜔, 𝑖_𝑑, and 𝑖_𝑞 are the normalized motor angular speed and the normalized quadrature-axis and direct-axis currents, respectively.

Table 1 

The denotation of the PMSM parameters.

Then, the new normalized model for the PMSM is rewritten as 𝑑𝜔𝑖𝑑𝑡=𝜎_𝑞−𝜔−𝑇_𝐿,𝑑𝑖_𝑞𝑑𝑡=−𝑖_𝑞−𝑖_𝑑𝜔+𝛾𝜔+𝑢_𝑞,𝑑𝑖_𝑑𝑑𝑡=−𝑖_𝑑+𝑖_𝑞𝜔+𝑢_𝑑, (3) where 𝛾=−𝜓_𝑟/(𝜅𝐿), 𝜎=𝐵𝜏/𝐽, 𝑇_𝐿=𝜏²𝑇_𝐿/𝐽, 𝑢_𝑞=𝑢_𝑞/(𝜅𝑅), 𝑢_𝑑=𝑢_𝑑/(𝜅𝑅), 𝑢_𝑞 and 𝑢_𝑑 denote the normalized quadrature-axis and direct-axis stator voltage, respectively, 𝑇_𝐿 presents the normalized load torque, and 𝜎 and 𝛾 are previously defined system parameter.

Figures 1 and 2 reveal the chaotic behavior of the PMSM for the situation of 𝜎=5.45, 𝛾=20, 𝑢_𝑞=𝑢_𝑑=0, 𝑇_𝐿=0, 𝜔(0)=−5, 𝑖_𝑞(0)=0.01, and 𝑖_𝑑(0)=20, in which it appears as an aperiodic, random, sudden, or intermittent morbid oscillation.

Figure 1 

Three-dimensional phase diagram with parameters 𝜎=5.45 and 𝛾=20.

(a) The angular speed

(b) The normalized 𝑞-axis current

(c) The normalized 𝑑-axis current

Figure 2 

The chaotic time series of the PMSM.

It is obvious that the model of the PMSM has high nonlinearity because of the coupling between the speed and the currents. In addition, the indeterminate system parameters 𝜎 and 𝛾 are enormously impacted by realistic conditions. So as to control efficiently the PMSM, 𝑢_𝑞 and 𝑢_𝑑 are used as the manipulated variables. Then, a nonlinear dynamics surface control approach based on RBF neural network is proposed to restrain the chaos, parameters variation, and external disturbance in the PMSM.

For the sake of simplicity, the following symbols are introduced: 𝑥₁=𝜔,𝑥₂=𝑖_𝑞,𝑥₃=𝑖_𝑑. (4)

Then, the mathematic model of the PMSM can be represented as follows: 𝑥̇₁=𝜎𝑥₂−𝜎𝑥₁−𝑇_𝐿+Δ₁,𝑥̇₂=−𝑥₂−𝑥₁𝑥₃+𝛾𝑥₁+𝑢_𝑞+Δ₂,𝑥̇₃=−𝑥₃+𝑥₁𝑥₂+𝑢_𝑑+Δ₃, (5) where Δ_𝑖, 𝑖=1--3, denote the external disturbance.

Assumption 1. The unknown disturbance terms Δ_𝑖 satisfy |Δ_𝑖(𝑥,𝑡)|<𝑑_𝑖, 𝑖 = 1–3, and the parameter 𝜎 is unknown and bounded. Moreover, it is assumed that |𝜎|≤𝐺.

Assumption 2. The desired trajectory 𝑦_𝑟 is continuous, and its first-order derivative 𝑦̇_𝑟 and second-order derivative 𝑦̈_𝑟 are bounded and available.

3. Controller Design

3.1. RBF Neural Network

The type of RBF neural network is considered as a two-layer network, which contains a hidden layer and an output layer. In this paper, the RBF neural network will be used to approximate the unknown continuous function 𝑓(𝑧)∶𝑅^𝑛→𝑅 as follows: 𝑓(𝑧)=𝜃^∗𝑇𝜉(𝑧), (6) where 𝑧∈Ω⊂𝑅^𝑛 is the input vector with 𝑛 being the neural network input dimension, 𝜃^∗=[𝜃^∗₁,𝜃^∗₂,…,𝜃^∗_𝑙]^𝑇∈𝑅^𝑙 is the weight vector, 𝑙>1 is the node number of neuron, and 𝜉(𝑧)=[𝜉₁(𝑧),𝜉₂(𝑧),…,𝜉_𝑛(𝑧)]^𝑇∈𝑅^𝑙 is a basic function vector with 𝜉_𝑖(𝑧) chosen as the commonly used Gaussian function in the following form: 𝜉_𝑖(𝑧)=exp−𝑧−𝜇_𝑖²2𝜎²_𝑖=exp−𝑧−𝜇_𝑖^𝑇𝑧−𝜇_𝑖2𝜎²_𝑖,𝑖=1,2,…,𝑙, (7) where 𝜇_𝑖=[𝜇_𝑖1,𝜇_𝑖2,…,𝜇_𝑖𝑛]^𝑇 is the center of the receptive field and 𝜎_𝑖 is the width of 𝜉_𝑖(𝑧).

For given scalar 𝜀>0, by choosing sufficiently large 𝑙, the RBF neural network can approximate any continuous function over a compact set Ω∈𝑅^𝑛 to arbitrary accuracy as follows: 𝑓(𝑧)=𝜃^𝑇𝜉(𝑧)+𝜀(𝑧),∀𝑧∈Ω∈𝑅^𝑛, (8) where 𝜀(𝑧) is the approximation error and 𝜃 is an unknown ideal constant weight vector, which is an artificial quantity required for analytical purpose. Typically, 𝜃 is chosen as the value of 𝜃^∗ that minimizes |𝜀(𝑧)|, for all 𝑧∈Ω; that is, 𝜃=argmin_{𝜃∗∈𝑅𝑛}sup_𝑧∈Ω𝑓(𝑧)−𝜃^∗𝑇𝜉(𝑧). (9)

Assumption 3. The approximation error 𝜀 is bounded, and it has positive constant 𝜀_𝑀 which satisfies |𝜀_𝑖|≤𝜀_𝑀,𝑖 = 1–3.

Assumption 4. There exists a positive and bounded constant 𝜌_𝑖 which satisfies |𝜀_𝑀+𝑑_𝑖|≤𝜌_𝑖.

3.2. The Controller of Dynamics Surface Control Approach Based on RBF Neural Network

In this section, the controller of dynamics surface control approach will be developed based on the minimum weights of RBF neural network. The design procedure consists of three steps. Then, the detail process will be given.

Step 1. The first dynamic surface is defined as 𝑆₁=𝑥₁−𝑦_𝑟. Then, the time derivative of 𝑆₁ can be obtained as follows: 𝑆̇₁=𝜎𝑥₂+𝑓₁+Δ₁−𝑦̇_𝑟, (10) where 𝑓₁=−𝜎𝑥₁−𝑇_𝐿.

The operating parameter 𝜎 is usually unknown due to the effect of the work environment. It is difficult for traditional methods to deal with the problem. In order to solve it, the adaptive technique is introduced to estimate the 𝜎, and the adaptive RBF neural network is used to approximate the uncertain item 𝑓₁ with little error. Therefore, for any given 𝜀₁, there exists a RBF neural network 𝜃^𝑇₁𝜉₁ such that 𝑓₁=𝜃^𝑇₁𝜉₁+𝜀₁, (11) where 𝜀₁ is the approximation error and satisfies |𝜀₁|≤𝜀_𝑀.

Substituting (11) into (10) yields the following: 𝑆̇₁≤𝜎𝑥₂+𝜃^𝑇₁𝜉₁+𝜌₁−𝑦̇_𝑟. (12)

The virtual control and related adaptive laws can be designed as follows: 𝛼₂=𝜎𝜎²+𝜂−𝑘₁𝑆₁−12𝑎²₁𝜆₁𝑆₁𝜉^𝑇₁𝜉₁−12𝑆₁𝜌²₁+𝑦̇_𝑟,̇𝜆₁=12𝑎²₁𝛾₁𝜉^𝑇₁𝜉₁𝑆²₁−𝑚₁𝜆₁,𝜎̇=Γ₁𝑆₁𝛼₂−𝑐₁𝜎, (13)(14)(15) where 𝜆₁=‖𝜃₁‖² belongs to a minimum weights of RBF neural network which greatly speeds up the solution speed, 𝑘₁, 𝑚₁, 𝑎₁, 𝛾₁, 𝑐₁, and Γ₁ are the design constant, and 𝜂 is a small positive constant.

Introduce variables 𝜆₁ and 𝜎 as follows: 𝜆₁=𝜆₁−𝜆₁,𝜎=𝜎−𝜎. (16)

Let 𝛼₂ be passed through a first-order filter with a time constant 𝜏₂ as follows: 𝜏₂𝛼̇_2𝑓+𝛼_2𝑓=𝛼₂,𝛼_2𝑓(0)=𝛼₂(0). (17)

Define the filter error as 𝑦₂=𝛼_2𝑓−𝛼₂. With (17) and 𝑦₂, it yields that 𝛼̇_2𝑓𝑦=−²𝜏₂. (18)

Then, the time derivative of 𝑦₂ can be obtained as follows: 𝑦̇₂𝑦=−²𝜏₂−𝜎𝜎²+𝜂−𝑘₁𝑆̇₁−̇𝜃^𝑇₁𝜉₁−𝜃^𝑇₁𝜕𝜉₁𝜕𝑥₁𝑥̇₁−12𝑆̇₁𝜌²₁−𝑆₁𝜌₁𝜕𝜌₁𝜕𝑥₁𝑥̇₁+𝑦̈_𝑟−𝜂−𝜎²𝜎̇𝜎²+𝜂²−𝑘₁𝑆₁−𝜃^𝑇₁𝜉₁−12𝑆₁𝜌²₁+𝑦̇_𝑟. (19)

It is obtained that 𝑦̇₂+𝑦₂𝜏₂≤𝐵₂𝑆₁,𝑆₂,𝑦₂,𝜆₁,𝜎,𝑦_𝑟,𝑦̇_𝑟,𝑦̈_𝑟. (20)

Using Young’s inequality, one has 𝑦₂𝑦̇₂𝑦≤−²²𝜏₂+𝑦²₂+14𝐵²₂, (21) where 𝐵₂(𝑆₁,𝑆₂,𝑦₂,𝜆₁,𝜎,𝑦_𝑟,𝑦̇_𝑟,𝑦̈_𝑟) is the continuous function.

Substituting (13)–(21) into (12), (12) becomes 𝑆̇₁≤𝜎𝑆₂+𝑦₂−𝜎𝛼₂+𝜎²𝜎²×+𝜂−𝑘₁𝑆₁−12𝑎²₁𝜆₁𝑆₁𝜉^𝑇₁𝜉₁−12𝑆₁𝜌²₁+𝑦̇_𝑟+𝜃^𝑇₁𝜉₁+𝜌₁−𝑦̇_𝑟. (22)

One has 𝑆₁𝑆̇₁≤𝜎𝑆²₁+14𝑆²₂−𝜂𝜎²×+𝜂−𝑘₁𝑆₁−12𝑎²₁𝜆₁𝑆₁𝜉^𝑇₁𝜉₁−12𝑆₁𝜌²₁+𝑦̇_𝑟𝑆₁+𝜎𝑆²₁+14𝑦²₂−𝜎𝛼₂𝑆₁−𝑘₁𝑆²₁−12𝑎²₁𝜆₁𝜉^𝑇₁𝜉₁𝑆²₁+12+𝑎²₁2≤2𝐺−𝑘₁𝑆²₁+14𝐺𝑆²₂+14𝐺𝑦²₂−𝜎𝛼₂𝑆₁−12𝑎²₁𝜆₁𝜉^𝑇₁𝜉₁𝑆²₁+12+𝑎²₁2. (23)

Consider the Lyapunov function candidate as follows: 𝑉₁=12𝑆²₁+𝑦²₂+1𝛾₁𝜆²₁+1Γ₁𝜎². (24)

Then, the time derivative of 𝑉₁ is obtained as follows: 𝑉̇₁≤2𝐺−𝑘₁𝑆²₁+14𝐺𝑆²₂+12+𝑎²₁2+−1𝜏₂1+1+4𝐺𝑦²₂+14𝐵²₂−12𝑚₁𝛾₁𝜆²₁−12𝑐₁𝜎²+12𝑚₁𝛾₁𝜆²₁+12𝑐₁𝜎². (25) For the terms −𝑐₁𝜎𝜎 and −(1/𝛾₁)𝑚₁𝜆₁𝜆₁, one has −𝑐₁𝜎𝜎≤−(1/2)𝑐₁𝜎²+(1/2)𝑐₁𝜎² and −1𝛾₁𝑚₁𝜆₁𝜆₁1=−𝛾₁𝑚₁𝜆₁𝜆₁+𝜆₁1≤−2𝑚₁𝛾₁𝜆²₁+12𝑚₁𝛾₁𝜆²₁. (26)

Step 2. The second dynamic surface is given as 𝑆₂=𝑥₂−𝛼_2𝑓. (27)

Then, differentiating 𝑆₂ gives 𝑆̇₂=𝑓₂+Δ₂+𝑢_𝑞−𝛼̇_2𝑓, (28) where 𝑓₂=−𝑥₁𝑥₃−𝑥₂+𝛾𝑥₁.

To facilitate engineering application, a minimum-weights-based RBF neural network will be employed to approximate the nonlinear function 𝑓₂ again. Therefore, there exists a RBF neural network system such that 𝑓₂=𝜃^𝑇₂𝜉₂+𝜀₂, (29) where 𝜀₂ is the approximation error and satisfies |𝜀₂|≤𝜀_𝑀.

Substituting (29) into (28), one has 𝑆̇₂≤𝜃^𝑇₂𝜉₂+𝑢_𝑞+𝜌₂−𝛼̇_2𝑓. (30)

Similarly, the relevant control law and adaptive law are provided in the following forms: 𝑢_𝑞=−𝑘₂𝑆₂−12𝑎²₂𝜆₂𝑆₂𝜉^𝑇₂𝜉₂−12𝑆₂𝜌²₂+𝛼̇_2𝑓,̇𝜆₂=12𝑎²₂𝛾₂𝜉^𝑇₂𝜉₂𝑆²₂−𝑚₂𝜆₂, (31)(32) where 𝑘₂, 𝑚₂, 𝑎₂, and 𝛾₂ are the design constant and 𝜆₂=‖𝜃₂‖².

With (31) and (32), (30) is written as follows: 𝑆̇₂≤−𝑘₂𝑆₂+𝜃^𝑇₂𝜉₂+𝜌₂−12𝑎²₂𝜆₂𝑆₂𝜉^𝑇₂𝜉₂−12𝑆₂𝜌²₂. (33)

One has 𝑆₂𝑆̇₂≤−𝑘₂𝑆²₂−12𝑎²₂𝜆₂𝑆²₂𝜉^𝑇₂𝜉₂+𝑎²₂2+12. (34)

Choose the Lyapunov function candidate as follows: 𝑉₂=12𝑆²₂+1𝛾₂𝜆²₂. (35) Then, the time derivative of 𝑉₂ is 𝑉̇₂≤−𝑘₂𝑆²₂+𝑎²₂2+12−12𝑚₂𝛾₂𝜆²₂+12𝑚₂𝛾₂𝜆²₂. (36) For the term −(1/𝛾₂)𝑚₂𝜆₂𝜆₂, one has −(1/𝛾₂)𝑚₂𝜆₂𝜆₂≤−(1/2)(𝑚₂/𝛾₂)𝜆²₂+(1/2)(𝑚₂/𝛾₂)𝜆²₂.

Step 3. Choose the last dynamic surface as 𝑆₃=𝑥₃. Then, the time derivative of 𝑆₃ is calculated as follows: 𝑆̇₃=𝑓₃+𝑢_𝑑+Δ₃, (37) where 𝑓₃=−𝑥₃+𝑥₁𝑥₂.

In the same way, there is a minimum-weights-based RBF neural network such that 𝑓₃=𝜃^𝑇₃𝜉₃+𝜀₃, (38) where 𝜀₃ is the approximation error and satisfies |𝜀₃|≤𝜀_𝑀.

Substituting (38) into (37) gives 𝑆̇₃≤𝜃^𝑇₃𝜉₃+𝜌₃+𝑢_𝑑. (39)

At the present stage, the control input is designed as follows: 𝑢_𝑑=−𝑘₃𝑆₃−12𝑎²₃𝜆₃𝑆₃𝜉^𝑇₃𝜉₃−12𝑆₃𝜌²₃, (40) where 𝑘₃ is the positive constant and 𝜆₃=‖𝜃₃‖².

According to the mention above, the adaptive law is chosen as follows: ̇𝜆₃=12𝑎²₃𝛾₃𝜉^𝑇₃𝜉₃𝑆²₃−𝑚₃𝜆₃, (41) where 𝑚₃, 𝑎₃, and 𝛾₃ are the design constant.

Similarly, (37) is given as follows: 𝑆̇₃≤𝜃^𝑇₃𝜉₃+𝜌₃−𝑘₃𝑆₃−12𝑎²₃𝜆₃𝑆₃𝜉^𝑇₃𝜉₃−12𝑆₃𝜌²₃. (42)

There exists 𝑆₃𝑆̇₃1≤−2𝑎²₃𝜆₃𝑆²₃𝜉^𝑇₃𝜉₃−𝑘₃𝑆²₃+𝑎²₃2+12. (43)

Choose the Lyapunov function candidate as follows: 𝑉₃=12𝑠²₃+1𝛾₃𝜆²₃. (44) Using the equality in (44), it can be verified easily that 𝑉̇₃≤−𝑘₃𝑆²₃+𝑎²₃2+12−12𝑚₃𝛾₃𝜆²₃+12𝑚₃𝛾₃𝜆²₃. (45) For the term −(1/𝛾₃)𝑚₃𝜆₃𝜆₃, one has −(1/𝛾₃)𝑚₃𝜆₃𝜆₃≤−(1/2)(𝑚₃/𝛾₃)𝜆²₃+(1/2)(𝑚₃/𝛾₃)𝜆²₃.

Up to now, the design procedure of proposed controller of the PMSM is completed. The proposed controller significantly reduces the computation complexity compared with traditional backstepping control and dynamics surface control. Based on previous procedure, the configuration of the proposed control system is depicted in Figure 3. The overall configuration consists of the PMSM with load, the space vector pulse width modulation (SVPWM), the voltage-source inverter, the power source rectifier, the automatic current regulator of the motor, the encoder used to detect speed and position, 𝑞-axis and 𝑑-axis controllers.

Figure 3 

Control schematic of the PMSM.

4. Stability Analysis

For any given 𝑝>0, the closed sets can be defined as follows: Π₁=𝑆₁,𝜆₁,𝜎,𝑦₂∶𝑆²₁+𝑦²₂+1𝛾₁𝜆²₁+1Γ₁𝜎²,Π≤2𝑝₂=𝑆₁,𝑆₂,𝜆₁,𝜆₂,𝜎,𝑦₂∶_2𝑖=1𝑆²_𝑖+𝑦²₂+²_𝑖=11𝛾_𝑖𝜆²_𝑖+1Γ₁𝜎²,Π≤2𝑝₃=𝑆₁,…,𝑆₃,𝜆₁,…,𝜆₃,𝜎,𝑦₂∶_3𝑖=1𝑆²_𝑖+𝑦²₂+³_𝑖=11𝛾_𝑖𝜆²_𝑖+1Γ₁𝜎²≤2𝑝. (46)

Theorem 5. Suppose that the control laws in (31) and (40) with adaptive laws (14), (15), (32), and (41) are applied to the PMSM; Assumptions 1–4 stand, and if there exists a positive constant 𝑝 that the initial condition satisfies 𝑉₃≤𝑝 and the design constants 𝑘_𝑖, 𝑎_𝑖, 𝛾_𝑖, 𝑚_𝑖, Γ₁, and 𝑐₁ are chosen rationally, then all the signals of the closed-loop system are semiglobally uniformly ultimately bounded, and the output tracking error converges to a neighborhood of zero.

Proof. Define the Lyapunov function candidate as follows: 1𝑉=2³_𝑖=1𝑆²_𝑖+𝑦²₂+³_𝑖=11𝛾_𝑖𝜆²_𝑖+1Γ₁𝜎². (47)
Consequently, one can obtain 𝑉̇≤2𝐺−𝑘₁𝑆²₁−𝑘₂−14𝐺𝑆²₂−𝑘₃𝑆²₃+−1𝜏₂1+1+4𝐺𝑦²₂−12𝑐₁𝜎²−12³_𝑖=1𝑚_𝑖𝛾_𝑖𝜆²_𝑖+12³_𝑖=1𝑚_𝑖𝛾_𝑖𝜆²_𝑖+12𝑐₁𝜎²+³_𝑖=1𝑎²_𝑖2+32+14𝐵²₂≤−2𝑎₀𝑉+𝑏, (48) where 𝑏=(1/2)∑³_𝑖=1(𝑚_𝑖/𝛾_𝑖)𝜆²_𝑖+(1/2)𝑐₁𝜎²+∑³_𝑖=1(𝑎²_𝑖/2)+(3/2)+(1/4)𝐵²₂.
Furthermore, (48) implies that (𝑡)≤𝑏0≤𝑉𝑎₀+𝑉(0)−𝑏𝑎₀𝑒^−𝑎0𝑡≤𝑏𝑎₀+𝑉(0). (49)

5. Performance Evaluation

The simulation is running to illustrate the effectiveness of the scheme presented in this paper under the assumption that the system parameters and nonlinear functions are uncertain. The initial conditions 𝑥₁(0)=𝑥₂(0)=1 and 𝑥₃(0)=1.5 are used in this section. The controller parameters are chosen as follows: 𝑘₁=35, 𝑘₂=𝑘₃=15, 𝛾₁=𝛾₂=𝛾₃=2, 𝑚₁=𝑚₂=𝑚₃=0.04, 𝑎₁=𝑎₂=𝑎₃=30, 𝜏₁=0.01, 𝜂=0.01, Γ₁=20, 𝑐₁=0.02, 𝜆₁(0)=𝜆₂(0)=𝜆₃(0)=0, and 𝜎(0)=0.7.

To take into account the disturbance, the corresponding expressions are given as follows: Δ₁=Δ₂=Δ₃=0.04×𝑥²₁cos³(),𝑇3𝑡_𝐿=2.0Nm,0≤𝑡≤43.5Nm,𝑡>4. (50)

The simulation results are shown in Figures 4(a)–4(f). Figures 4(a) and 4(b) explicitly illustrate that the state error of angular speed of the PMSM is gradually converged to zero within a short time. Meanwhile, these pictures show that the system successfully escapes from the chaotic behavior within 0.1 s and tracks the given trajectory with a great performance in spite of uncertainty, nonlinearity, and external disturbance. In Figures 4(c)–4(f), three kinds of curves basically overlap without disturbance on the whole time. Obviously, these pictures show the performance when the system parameters 𝜎 and 𝛾 have a perturbation. Furthermore, the indicators of the PMSM can still converge quickly when the model suffers from the disturbance. From the simulation results, it can clearly be seen that the proposed controller guarantees the boundedness of all the signals in the closed-loop system and also achieves the good tracking performance.

(a) The trajectory tracking

(b) The rotor velocity tracking error

(c) 𝑞-axis voltage

(d) 𝑑-axis voltage

(e) The periodic orbit

(f) The 𝑑-axis current

Figure 4 

The robustness analysis with 0.5sin(4𝑡+1/2𝜋) + 0.5sin(2𝑡+1/2𝜋).

6. Conclusion

In this paper, an adaptive dynamic surface control method of chaos is applied to the permanent magnet synchronous motor based on the minimum weights of RBF neural network. The proposed controller guarantees the boundedness of all the signals in the closed-loop system, while the tracking error eventually converges to a small neighborhood of the origin. Moreover, the suggested controller contains minimum weights of RBF neural network. This makes the design scheme easier to be implemented in practical applications. Simulation results are given to show the effectiveness and robustness of the proposed controller. Finally, some potential future research works are pointed out, such as the recursive filtering for time-varying nonlinear systems and sliding mode design for time-invariant nonlinear systems.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This Project is supported by the National Natural Science Foundation of China (Grant no. 61203080) and Scientific and Technological Research Program of Chongqing Municipal Education Commission (Grant no. KJ133202). The author also gratefully acknowledges the helpful comments and suggestions of the reviewers.